{
  "id": "1705.07048",
  "version": "v1",
  "published": "2017-05-19T15:22:38.000Z",
  "updated": "2017-05-19T15:22:38.000Z",
  "title": "Linear regression without correspondence",
  "authors": [
    "Daniel Hsu",
    "Kevin Shi",
    "Xiaorui Sun"
  ],
  "categories": [
    "cs.LG",
    "math.ST",
    "stat.ML",
    "stat.TH"
  ],
  "abstract": "This article considers algorithmic and statistical aspects of linear regression when the correspondence between the covariates and the responses is unknown. First, a fully polynomial-time approximation scheme is given for the natural least squares optimization problem in any constant dimension. Next, in an average-case and noise-free setting where the responses exactly correspond to a linear function of i.i.d. draws from a standard multivariate normal distribution, an efficient algorithm based on lattice basis reduction is shown to exactly recover the unknown linear function in arbitrary dimension. Finally, lower bounds on the signal-to-noise ratio are established for approximate recovery of the unknown linear function by any estimator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-19T15:22:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear regression",
      "unknown linear function",
      "correspondence",
      "standard multivariate normal distribution",
      "squares optimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}